Intermediate mathematics
:See also: Category:Foundations It has been known since the time of Euclid that all of geometry can be derived from a handful of objects (points, lines...), a few actions on those objects, and a small number of axioms. Every field of science likewise can be reduced to a small set of objects, actions, and rules. Math itself is not a single field but rather a constellation of related fields. One way in which new fields are created is by the process of generalization. A generalization is the formulation of general concepts from specific instances by abstracting common properties. Generalization is the process of identifying the parts of a whole, as belonging to the whole.Wikipedia:Generalization Foreword Mathematical notation can be extremely intimidating. Wikipedia is full of articles with page after page of indecipherable text. At first glance this article might appear to be the same. I want to assure the reader that every effort has been made to simplify everything as much as possible and to provide all relevant information or, at very least, to make such information easy to find. The following has been assembled from countless small pieces gathered from throughout the world wide web. I cant guarantee that there are no errors in it. Please report any errors or omissions on this articles talk page. Euclids "common notions" :From Wikipedia:Euclidean geometry Things that do not differ from one another are equal to one another Things that are equal to the same thing are also equal to one another | then a=c |} If equals are added to equals, then the wholes are equal | then a+c=b+d |} If equals are subtracted from equals, then the remainders are equal | then a-c=b-d |} The whole is greater than the part. Numbers :See: Peano axioms The basis of all of mathematics is the "Next" function (see Graph theory). Next(0)=1, Next(1)=2, Next(2)=3, Next(3)=4. This defines the Natural numbers (denoted \mathbb{N}_0 ). :These have the convenient property of being transitive. That means that if a1-3=x for which there is no answer among natural numbers. To provide an answer mathematicians generalize to the set of all integers (denoted \mathbb{Z} ) which includes negative integers. :The Additive identity is zero because x + 0 = x. : (denoted \mathcal O_\mathbb{Q} ) over the field of rational numbers. Ring is defined below. :The study of integers is called Number theory. :: a \mid b means a divides b. :: a \nmid b means a does not divide b. :: p^a \mid\mid n means pa exactly divides n (i.e. pa divides n but pa+1 does not). Multiplication is defined as repeated addition, and its inverse is division. But this leads to equations like 3/2=x for which there is no answer. The solution is to generalize to the set of rational numbers (denoted \mathbb{Q} ). Any number which isnt rational is irrational. :Rational numbers form a field. Field is defined below. :The Multiplicative identity is one because x * 1 = x. :Division by zero is undefined and undefinable. 1/0 exists nowhere on the number line nor anywhere on the complex plane. It does, however, exist on the where it is surprisingly well behaved. See also and L'Hôpital's rule. :(Addition and multiplication are fast but division is slow even for computers.) Exponentiation is defined as repeated multiplication, and its inverses are roots and logarithms. But this leads to multiple equations with no solutions: :Equations like \sqrt{2}=x. The solution is to generalize to the set of algebraic numbers (denoted \mathbb{A} ). See also . ::Equations like 2^{\sqrt{2}}=x The solution (because x is transcendental) is to generalize to the set of Real numbers (denoted \mathbb{R} ). :Equations like \sqrt{-1}=x and e^x=-1. The solution is to generalize to the set of complex numbers (denoted \mathbb{C} ) by defining i = sqrt(-1). A single complex number z=a+bi consists of a real part a'' and an imaginary part ''bi. ::The Complex conjugate of the complex number z=a+bi is \bar{z}=a-bi. (Not to be confused with the dual of a vector.) ::: \frac{a+bi}{c+di}\cdot\frac{c-di}{c-di}=\left( \frac{ac+bd}{c^2+d^2} \right) + \left( \frac{bc-ad}{c^2+d^2} \right)i :0^0 = 1. See Empty product. Tetration is defined as repeated exponentiation and its inverses are called super-root and super-logarithm. : \begin{matrix} {}^{b}a & = & \underbrace{a^{a^ }}}} & = & a\uparrow\uparrow b = & \underbrace{a\uparrow (a\uparrow(\dots\uparrow a))} & \\ & & b\mbox{ copies of }a & & & b\mbox{ copies of }a \end{matrix} Imaginary numbers (denoted \mathbb{I} ) often occur in equations involving change with respect to time. If friction is resistance to motion then imaginary friction would be resistance to change of motion wrt time. (In other words, imaginary friction would be mass.) In fact, in the equation for the Spacetime interval (given below), time itself is an imaginary quantity. , in stereographic projection, in double rotation]] Complex numbers can be used to represent and perform rotations but only in 2 dimensions. Hypercomplex numbers like quaternions (denoted \mathbb{H} ), octonions (denoted \mathbb{O} ), and (denoted \mathbb{S} ) are one way to generalize complex numbers to some (but not all) higher dimensions. Tensors, on the other hand, can be used in any number of dimensions to represent and perform rotations and other linear transformations. See Visualization of Tensor multiplication. Rotations in n dimensions are called SO(n). In 4 spatial dimensions a rigid object can . :Any affine transformation is equivalent to a linear transformation followed by a translation of the origin. (The origin is always a fixed point for any linear transformation.) "Translation" is just a fancy word for "move". When a quantity, like the charge of a single electron, becomes so small that it is insignificant we, quite justifiably, treat it as though it were zero. A quantity that can be treated as though it were zero, even though it very definitely is not, is called infinitesimal. If q is a finite ( q \cdot 1 ) amount of charge then using Leibniz's notation dq would be an infinitesimal ( q \cdot 1/\infty ) amount of charge. See Differential Likewise when a quantity becomes so large that a regular finite quantity becomes insignificant then we call it infinite. We would say that the mass of the ocean is infinite ( M \cdot \infty ) . But compared to the mass of the Milky Way galaxy our ocean is insignificant. So we would say the mass of the Galaxy is doubly infinite ( M \cdot \infty^2 ) . Infinity and the infinitesimal are called Hyperreal numbers (denoted {}^*\mathbb{R} ). Hyperreals behave, in every way, exactly like real numbers. For example, 2 \cdot \infty is exactly twice as big as \infty. In reality, the mass of the ocean is a real number so it is hardly surprising that it behaves like one. See and Intervals :[-2,5[ or denotes the interval from -2 to 5, including -2 but excluding 5. :[3..7 denotes all integers from 3 to 7. :The set of all reals is unbounded at both ends. :An open interval does not include its endpoints. :: is a property that generalizes the notion of a subset being closed and bounded. :The unit interval is the closed interval 0,1. It is often denoted I. :The unit square is a square whose sides have length 1. Often, "the" unit square refers specifically to the square in the Cartesian plane with corners at the four points (0, 0), (1, 0), (0, 1), and (1, 1). :The unit disk in the complex plane is the set of all complex numbers of absolute value less than one and is often denoted \mathbb {D} Back to top Vectors :See also: , , , , and Linear algebra The one dimensional number line can be generalized to a multidimensional Cartesian coordinate system thereby creating multidimensional math (i.e. geometry). : \mathbb{R}^3 is the Cartesian product \mathbb{R} \times \mathbb{R} \times \mathbb{R}. : \mathbb{R}^\infty = \mathbb{R}^\mathbb{N} : \mathbb{C}^3 is the Cartesian product \mathbb{C} \times \mathbb{C} \times \mathbb{C} A vector space is a with vector addition and scalar multiplication (multiplication of a vector and a scalar belonging to a field. :If {\mathbf e_1} , {\mathbf e_2} , {\mathbf e_3} are orthogonal unit ). :and {\mathbf u} , {\mathbf v} , {\mathbf x} are arbitrary vectors then we can (and usually do) write: ::'' \mathbf{u} = u_1 \mathbf{e_1} + u_2 \mathbf{e_2} + u_3 \mathbf{e_3} = \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix} '' ::'' \mathbf{v} = v_1 \mathbf{e_1} + v_2 \mathbf{e_2} + v_3 \mathbf{e_3} = \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix} '' ::'' \mathbf{x} = x_1 \mathbf{e_1} + x_2 \mathbf{e_2} + x_3 \mathbf{e_3} = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} '' :A generalizes a vector space by allowing multiplication of a vector and a scalar belonging to a ring. Coordinate systems define the length of vectors parallel to one of the axes but leave all other lengths undefined. This concept of "length" which only works for certain vectors is generalized as the " " which works for all vectors. The norm of vector \mathbf{v} is denoted \|\mathbf{v}\|. The double bars are used to avoid confusion with the absolute value of the function. A is a that is also a complete metric space (there are no points missing from it). : (called L''1'' norm. See .) :: \|\mathbf{v}\| = v_1 + v_2 + v_3 :In Euclidean space the norm (called ''L'2'' norm) doesnt depend on the choice of coordinate system therefore rigid objects can rotate. See proof of the Pythagorean theorem to the right. See also Lebesgue measure. :: \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} :In Minkowski space the Spacetime interval is :: \|s\| = \sqrt{x^2 + y^2 + z^2 + (cti)^2} :In the most common norm of an n dimensional vector is obtained by treating it as though it were a regular real valued 2n dimensional vector in Euclidean space :: \left\| \boldsymbol{z} \right\| = \sqrt{z_1 \bar z_1 + \cdots + z_n \bar z_n} A manifold \mathbf{M} is a type of topological space in which each point has an infinitely small neighbourhood that is homeomorphic to Euclidean space. A manifold is locally, but not globally, Euclidean. :A \mathbf{T}_p \mathbf{M} is the set of all vectors tangent to \mathbf{M} at point p. :Informally, a \mathbf{TM} (red cylinder in image to the right) on a differentiable manifold \mathbf{M} (blue circle) is obtained by joining all the (red lines) together in a smooth and non-overlapping manner.Wikipedia:Tangent bundle The tangent bundle always has twice as many dimensions as the original manifold. ::A is the same thing minus the requirement that it be tangent. ::A vector bundle is a fiber bundle whose fibers are vector spaces. :::A is a generalization of a vector bundle. :The ( ) of a differentiable manifold is obtained by joining all the (pseudovector spaces). The cotangent bundle always has twice as many dimensions as the original manifold. Sections of that bundle are known as differential one-forms. A is a group that is also a differentiable manifold. :a (See ) is a local or linearized version of a Lie group. ::The Lie derivative generalizes the Lie bracket which generalizes the wedge product which is a generalization of the cross product which only works in 3 dimensions. The cross product is neither commutative nor associative and therefore doesnt form a field or even a ring (see below). :See Back to top Multiplication of vectors Multiplication can be generalized to allow for multiplication of vectors in 3 different ways: Dot product (a Scalar): \mathbf{u} \bullet \mathbf{v} = \| \mathbf{u} \|\ \| \mathbf{v}\| \cos(\theta) = u_1 v_1 + u_2 v_2 + u_3 v_3 : \mathbf{u}\bullet\mathbf{v} = \begin{bmatrix}u_1 \mathbf{e_1} \\ u_2 \mathbf{e_2} \\ u_3 \mathbf{e_3} \end{bmatrix} \begin{bmatrix}v_1 \mathbf{e_1} & v_2 \mathbf{e_2} & v_3 \mathbf{e_3} \end{bmatrix} = \begin{bmatrix}u_1 v_1 + u_2 v_2 + u_3 v_3 \end{bmatrix} :Strangely, only parallel components multiply. In Euclidean space \|\mathbf{v}\|^2 = \mathbf{v}\bullet\mathbf{v}= Q(\mathbf{x}). The dot product of a rank n tensor and a rank m tensor results in a rank n-m tensor. ::The dot product can be generalized to the bilinear form \beta(\mathbf{u,v}) = u^T Av = scalar where A is an (0,2) tensor. (For the dot product A is the identity tensor). Its associated is Q(\mathbf{x}) = \beta(\mathbf{x,x}). Two vectors are orthogonal if \beta(\mathbf{u,v}) = 0. A bilinear form is symmetric if \beta(\mathbf{u,v}) = \beta(\mathbf{v,u}) ::The dot product can also be generalized to the inner product (a sesquilinear form) \langle u,v\rangle=\bar{u}\bullet v=\overline{\langle v,u\rangle}=\overline{\bar{v} \bullet u}. The 2 vectors are called "bra" and "ket". :::A is an inner product space that is also a Complete metric space. Outer product (a tensor called a ): \mathbf{u} \otimes \mathbf{v}. :As one would expect, every component of one vector multipies with every component of the other vector. :Taking the dot product of u'⊗'v and any vector x''' (See Visualization of Tensor multiplication) causes the components of '''x not pointing in the direction of v''' to become zero. What remains is then rotated from '''v to u'''. :A rotation matrix can be constructed by summing three outer products. The first two sum to form a bivector. The third one rotates the axis of rotation zero degrees. \mathbf{e}_1 \otimes \mathbf{e}_2 - \mathbf{e}_2 \otimes \mathbf{e}_1 + \mathbf{e}_3 \otimes \mathbf{e}_3 : \mathbf{e}_1 \otimes \mathbf{e}_2 \bullet \mathbf{e}_2 = \mathbf{e}_1 ::The Tensor product generalizes the outer product. The tensor product of a rank n tensor and a rank m tensor results in a rank n+m tensor. Wedge product (a simple bivector): \mathbf{u} \wedge \mathbf{v} = \mathbf{u} \otimes \mathbf{v} - \mathbf{v} \otimes \mathbf{u} = \overline{\mathbf{v}} :The wedge product is also called the exterior product (sometimes mistakenly called the outer product). The term "exterior" comes from the exterior product of two vectors not being a vector. Just as a vector has length and direction so a bivector has an area and an orientation. In three dimensions \mathbf{u} \wedge \mathbf{v} is a pseudovector and its dual is the cross product. \overline{\mathbf{u} \wedge \mathbf{v}} = \mathbf{u} \times \mathbf{v} : (\mathbf{a} \wedge \mathbf{b}) \wedge \mathbf{c} = \mathbf{a} \wedge (\mathbf{b} \wedge \mathbf{c}) : (\mathbf{a} + \mathbf{b}) \wedge (\mathbf{c} + \mathbf{d}) = (\mathbf{a} \wedge \mathbf{c}) + (\mathbf{a} \wedge \mathbf{d}) + (\mathbf{b} \wedge \mathbf{c}) + (\mathbf{b} \wedge \mathbf{d}) : \mathbf{u} \wedge \mathbf{v} = -\mathbf{v} \wedge \mathbf{u} : \mathbf{u} \wedge \mathbf{u} = 0 : ::The a∧b∧c is a trivector which is a rank-3 tensor. ::In 3 dimensions a trivector is a pseudoscalar so in 3 dimensions every trivector can be represented as a scalar times the unit trivector. See Levi-Civita symbol ::: \mathbf{a}\cdot(\mathbf{b}\times \mathbf{c}) \cdot \mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3 :The Matrix commutator generalizes the wedge product. :: A_2 = A_1A_2 - A_2A_1 The dual of '''a is ā': : \overline{\mathbf{a}} \quad\stackrel{\rm def}{=} \quad\begin{bmatrix}\,\,0&\!-a_3&\,\,\,a_2\\\,\,\,a_3&0&\!-a_1\\\!-a_2&\,\,a_1&\,\,0\end{bmatrix} Back to top Trigonometry The law of cosines reduces to the Pythagorean theorem when gamma=90 degrees : c^2 = a^2 + b^2 - 2ab\cos\gamma, The 'law of sines (also known as the "sine rule") for an arbitrary triangle states: : \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = \frac{abc}{2\Delta}, where \Delta is the area of the triangle : \mbox{Area} = \Delta = \frac{1}{2}a b\sin C. The law of tangents: : \frac{a-b}{a+b}=\frac{\tan\left\tfrac{1}{2}(A-B)\right}{\tan\left\tfrac{1}{2}(A+B)\right} Euler's formula, which states that e^{ix} = \cos x + i \sin x , produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i'': : \sin x = \frac{e^{ix} - e^{-ix}}{2i}, \qquad \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \tan x = \frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}. Right triangles A right triangle is a triangle with gamma=90 degrees. For small values of x, sin x ≈ x. (If x is in radians). Back to top Polynomials :See , , and :From Wikipedia:Polynomial A polynomial can always be written in the form : polynomial = Z(y) = a_0 + a_1 y + a_2 y^2 + \dotsb + a_{n-1}y^{n-1} + a_n y^n where a_0, \ldots, a_n are constants called coefficients and ''n is the degree of the polynomial. :A is a polynomial of degree one. Each individual is the product of the and a variable raised to a nonnegative integer power. :A has only one term. :A has 2 terms. : :Every single-variable, degree n polynomial with complex coefficients has exactly n complex roots. However, some or even all of the roots might be the same number. A root (or zero) of a function is a value of y for which Z(y)=0. :: Z(y) = a_n(y - z_1)(y - z_2)\dotsb(y - z_n) :: The polynomial remainder theorem states that the remainder of the division of a polynomial Z(y) by the linear polynomial y-a is equal to Z(a). See . Determining the value at Z(a) is sometimes easier if we use ( ) by writing the polynomial in the form : Z(y) = a_0 + y(a_1 + y(a_2 + \cdots + y(a_{n-1} + y(a_n)))). A is a one variable polynomial in which the leading coefficient is equal to 1. : a_0 + a_1y + a_2y^2 + \cdots + a_{n-1}y^{n-1} + 1y^n A is a function of the form : f(y) = k{(y - z_1)(y - z_2)\dotsb(y - z_n) \over (y - p_1)(y - p_2)\dotsb(y - p_m)} = {Z(y) \over P(y)} It has n zeros and m poles. A pole is a value of y for which |f(y)| = infinity. :Given two polynomials Z(y) and P(y) = (y-p_1)(y-p_2) \cdots (y-p_m) , where the p''i'' are distinct constants and deg Z'' < ''m, partial fractions are generally obtained by supposing that :: \frac{Z(y)}{P(y)} = \frac{c_1}{y-p_1} + \frac{c_2}{y-p_2} + \cdots + \frac{c_n}{y-p_m} :and solving for the c''i'' constants, by substitution, by of terms involving the powers of y'', or otherwise. (This is a variant of the .)Wikipedia:Partial fraction decomposition A is given by : \sum_{y=0} c_y where c0=1 and {c_{y+1} \over c_y} = {Z(y) \over P(y)} = f(y) The function f(y) has n zeros and m poles. : , or hypergeometric q-series, are generalizations of generalized hypergeometric series.Wikipedia:Basic hypergeometric series ::Roughly speaking a of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1Wikipedia:q-analog ::We define the ''q-analog of n'', also known as the 'q''-bracket' or q-number of n'', to be ::: n_q=\frac{1-q^n}{1-q} = 1 + q + q^2 + \ldots + q^{n - 1} ::one may define the ''q-analog of the factorial, known as the , by ::: n_q! = 1_q \cdot 2_q \cdots n-1_q \cdot n_q : are generalizations of basic hypergeometric series. ::An elliptic function is a meromorphic function that is periodic in two directions. A is given by : F(x) = {}_nF_m(z_1,...z_n;p_1,...p_m;x) = \sum_{y=0} c_y x^y So for ex (see below) we have: : c_y = \frac{1}{y!}, \qquad \frac{c_{y+1}}{c_y} = \frac{1}{y+1}. Back to top Integration and differentiation :See also: Hyperreal number and Implicit differentiation The integral (antiderivative) is a generalization of multiplication. :For example: a unit mass dropped from point x2 to point x1 will release energy. The usual equation is is a simple multiplication: :: gravity \bullet (x_2 - x_1) = energy :But that equation cant be used if the strength of gravity is itself a function of x. The strength of gravity at x1 would be different than it is at x2. And in reality gravity really does depend on x (x is the distance from the center of the earth): :: gravity(x) = 1/x^2 (See inverse-square law.) :However, the corresponding Definite integral is easily solved: :: \int_{x_1}^{x_2} gravity(x) \cdot dx : :The integral of a function is equal to the area under the curve. When the "curve" is a constant (in other words, k•x0) then the integral reduces to ordinary multiplication. :The integral of 1/x is ln(x). See natural log The derivative is a generalization of division. The derivative of the integral of f(x) is just f(x). The derivative of f(x)=k•xy is : f'(x) = {df \over dx} = {d(k \cdot x^y) \over dx} \quad = \quad k \cdot {d(x^y) \over dx} \quad = \quad k \cdot y \cdot x^{y-1} The derivative of a function at any point is equal to the slope of the function at that point. The derivative of a constant (k•x0) is therefore zero. The equation of the line tangent to a function at point a is : y(x) = f(a) + f'(a)(x-a) The Lipschitz constant of a function is a real number for which the absolute value of the slope of the function at every point is not greater than this real number. .]] of a mass on a spring is a second-order linear ordinary differential equation. : F_\mathrm{net} = m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx, where is the inertial mass, is its displacement from the equilibrium, and is the spring constant. Solving for x produces : x(t) = A\cos\left(\omega t - \varphi\right), is the amplitude (maximum displacement from the equilibrium position), \omega = 2\pi f = \sqrt{k/m} is the angular frequency, and is the phase. Energy passes back and forth between the potential energy in the spring and the kinetic energy of the mass. The important thing to note here is that the frequency of the oscillation depends only on the mass and the stiffness of the spring and is totally independent of the amplitude. That is the defining characteristic of resonance. states that the sum of the emfs in any closed loop of any electronic circuit is equal to the sum of the in that loop.Wikipedia:RLC circuit : V(t) = V_R + V_L + V_C is the voltage, is the resistance, is the inductance, is the capacitance. : V(t) = RI(t) + L \frac{dI(t)}{dt} + \frac{1}{C} \int_{0}^t I(\tau)\, d\tau = dQ/dt is the current. It makes no difference whether the current is a small number of charges moving very fast or a large number of charges moving slowly. In reality . If V(t)=0 then the only solution to the equation is the transient response which is a rapidly decaying sine wave with the same frequency as the resonant frequency of the circuit. :Like a mass (inductance) on a spring (capacitance) the circuit will resonate at one frequency. Energy passes back and forth between the capacitor and the inductor with some loss as it passes through the resistor. If V(t)=sin(t) from -∞ to +∞ then the only solution is a sine wave with the same frequency as V(t) but with a different amplitude and phase. If V(t) is zero until t=0 and then equals sin(t) then I(t) will be zero until t=0 after which it will consist of the steady state response plus a transient response. If we know the value of a smooth function at x=0 (smooth means all its derivatives are continuous) and we know the value of all of its derivatives at x=0 then we can determine the value at any other point x'' by using the Maclaurin series. ("!" means factorial) : a_0 x^0 + a_1 x^1 + a_2 x^2 + a_3 x^3 + \cdots :where a_n = {f^{(n)}(0) \over n!} The proof of this is actually quite simple. Plugging in a value of ''x=0 causes all terms but the first to become zero. So, assuming that such a function exists, a0 must be the value of the function at x=0. Simply differentiate the function and repeat for the next term. And so on. :The Taylor series generalizes this formula. :: f(z)=\sum_{k=0}^\infty \alpha_k (z-z_0)^k :::An analytic function is a function whose Taylor series converges for every z0 in its domain; analytic functions are infinitely differentiable. ::::Any vector g'' = (''z''0, α0, α1, ...) is a if it represents a power series of an analytic function around ''z''0 with some radius of convergence ''r > 0. The set of germs \mathcal G is a Riemann surface. Riemann surfaces are the objects on which multi-valued functions become single-valued. :::::A of \mathcal G (i.e., an equivalence class) is called a . We can easily determine the Maclaurin series expansion of the exponential function e^x because it is equal to its own derivative. y = ex x = ln(y) dy/dx = y = ex dy/y = dx ∫ (1/y)dy = ∫ dx = x = ln(y) : e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = {x^0 \over 0!} + {x^1 \over 1!} + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots And cos(x) and sin(x) : \cos x = \frac{x^0}{0!} - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots : \sin x = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots It then follows that e^{ix}=\cos x+i\sin x=\operatorname{cis} x and therefore e^{i \pi}=-1 + i\cdot0 :x'' is the angle in . The Maclaurin series cant be used for a discontinuous function (see ) like a square wave because it is not differentiable but remarkably we can use the Fourier series to expand it or any other periodic function into an infinite sum of sine waves each of which is fully differentiable! : f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty \lefta_n\cos\left(nt\right)+b_n\sin\left(nt\right)\right : a_n = \frac{2}{p}\int_{t_0}^{t_p} f(t)\cdot \cos\left(\tfrac{2\pi nt}{p}\right)\ dt : b_n = \frac{2}{p}\int_{t_0}^{t_p} f(t)\cdot \sin\left(\tfrac{2\pi nt}{p}\right)\ dt ::The reason this works is because sine and cosine are . \langle sin,cos\rangle=0. That means that multiplying any 2 sine waves of frequency n and frequency m and integrating over one period will always equal zero unless n=m. See the graph of sin2(x) to the right. ::: 2 \sin mx \sin nx = \cos (m - n)x - \cos (m+n) x, ::::See ::And of course ∫ fn*(f1+f2+f3+...) = ∫ (fn*f1) + ∫ (fn*f2) + ∫ (fn*f3) +... :The complex form of the Fourier series uses complex exponentials instead of sine and cosine and uses both positive and negative frequencies (clockwise and counter clockwise) whose imaginary parts cancel. The complex coefficients encode both amplitude and phase and are complex conjugates of each other. ::: F(\nu) = \mathcal{F}\{f\} = \int_{\mathbb{R}^n} f(x) e^{-2 \pi i x\cdot\nu} \, \mathrm{d}x :::where the ''dot between x'' and ''ν indicates the inner product of R''n''. :A 2 dimensional Fourier series is used in video compression. :A can be computed very efficiently by a . :In mathematical analysis, many generalizations of Fourier series have proven to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space.Wikipedia:Generalized Fourier series : are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via Fourier series.Wikipedia:Spherical harmonics Spherical harmonics are for SO(3). See Laplace series. ::Every continuous function in the function space can be represented as a of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. Every quadratic polynomial can be written as a''1+''bt+''ct''2, that is, as a linear combination of the basis functions 1, t'', and ''t''2. Fourier transforms generalize Fourier series to nonperiodic functions like a single pulse of a square wave. The more localized in the time domain (the shorter the pulse) the more the Fourier transform is spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The Fourier transform of the Dirac delta function gives G(f)=1 : G(\omega)=\mathcal{F}\{f(t)\}=\int_{-\infty}^\infty f(t) e^{-i\omega t}dt :Laplace transforms generalize Fourier transforms to complex frequency s=\sigma+i\omega . Complex frequency includes a term corresponding to the amount of damping. See http://www.dspguide.com/CH32.PDF. :: F(s)=\mathcal{L}\{f(t)\}=\int_0^\infty f(t) e^{-\sigma t}e^{-i \omega t}dt ::: \mathcal{L}\{ \delta(t-a) \} = e^{-as} , (assuming ''a > 0) ::: \mathcal{L}\{e^{at} \}= \frac{1}{s - a} :The inverse Laplace transform is given by :: f(t) = \mathcal{L}^{-1} \{F\} = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}F(s)e^{st}\,ds, ::where the integration is done along the vertical line Re(s'') = ''γ in the complex plane such that γ'' is greater than the real part of all of ''F(s'') and ''F(s'') is bounded on the line, for example if contour path is in the . If all singularities are in the left half-plane, or ''F(s'') is an , then ''γ can be set to zero and the above inverse integral formula becomes identical to the .Wikipedia:Inverse Laplace transform :Integral transforms generalize Fourier transforms to other kernals (besides sine and cosine) ::Cauchy kernel = \frac{1}{\zeta-x} \quad \text{or} \quad \frac{1}{2\pi i} \cdot \frac{1}{\zeta-x} ::Hilbert kernel = cot\frac{\theta-t}{2} ::Poisson Kernel: :::For the ball of radius r, B_{r} , in R'n, the Poisson kernel takes the form: :::: P(x,\zeta) = \frac{r^2-|x|^2}{r} \cdot \frac{1} :Harmonic functions are real analogues to holomorphic functions. ::All harmonic functions are analytic, i.e. they can be locally expressed as power series. :::This is a general fact about , of which the Laplacian is a major example. ::The value of a harmonic function at any point inside a disk is a weighted average of the value of the function on the boundary of the disk. ::: Pu(x) = \int_S u(\zeta)P(x,\zeta)d\sigma(\zeta).\, :::The gives different weight to different points on the boundary except when x=0. The value at the center of the disk (x=0) equals the average of the equally weighted values on the boundary. All locally integrable functions satisfying the mean-value property are both infinitely differentiable and harmonic. :::The kernel itself appears to simply be 1/r^n shifted to the point x and multiplied by different constants. Green's theorem for functions in the complex plane : Cauchy's integral theorem for any holomorphic function f : : \oint f(z) \, dz = \iint \left( {\color{blue} \nabla \times f} + i {\color{red} \nabla \cdot f} \right) \, dx \, dy = 0 :(For real vector fields curl + divergence would be bivector + scalar. See Clifford algebra.) Computing the residue of a monomialWikipedia:Residue (complex analysis) : \begin{align} \oint_C z^k dz = \int_0^{2\pi} e^{i\theta k} \cdot i e^{i\theta} d\theta = \int_0^{2\pi} i e^{i\theta(k+1)} d\theta = \begin{cases} 2\pi i & \text{if } k = -1 \\ 0 & \text{otherwise} \end{cases} \end{align} :where C is the circle with radius 1 therefore z \to e^{i\theta} and dz \to d(e^{i\theta}) = ie^{i\theta}d\theta Cauchy's integral formula states that the value of a holomorphic scalar potential function within a disc is determined entirely by the values of the function on the boundary of the disc. Divergence can be nonzero outside the disc. Cauchy's integral formula can be generalized to more than two dimensions. : f(a)=\dfrac{1}{2\pi i}\oint_\gamma f(z)\frac{1}{z-a}dz For a unit circle: : f(z_0)=\dfrac{1}{2\pi}\oint_0^{2\pi}f(e^{i\theta})\frac{1-|z_0^2|} A 'morphism is exactly the same as a function but in Category theory every morphism has an inverse which is allowed to have more than one value or no value at all. consist of: :Objects (usually Sets) :Morphisms (usually maps) possessing: ::one source object (domain) ::one target object (codomain) a morphism is represented by an arrow: : f(x)=y is written f : x \to y where x is in X and y is in Y. : g(y)=z is written g : y \to z where y is in Y and z is in Z. The of y is z. The (or ) of z is the set of all y whose image is z and is denoted g^{-1}z A space Y is a (a fiber bundle) of space Z if the map g : y \to z is locally homeomorphic. :A covering space is a if it is . ::The concept of a universal cover was first developed to define a natural domain for the of an analytic function. :::The general theory of analytic continuation and its generalizations are known as . ::::The set of can be considered to be the analytic continuation of an analytic function. A topological space is if no part of it is disconnected. A space is if there are no holes passing all the way through it (therefore any loop can be shrunk to a point) :See Composition of morphisms: : g(f(x)) is written g \circ f ::f is the of g ::f is the of g \circ f ::? is the of ? A is a map from one set to another of the same type which preserves the operations of the algebraic structure: : f(x \bullet y) = f(x) \bullet f(y) : f(x + y) = f(x) + f(y) See ::A is a homomorphism with a domain in one category and a codomain in another. :A from (G'', ∗) to (''H, ·) is a h'' : ''G → H'' such that :: h(u*v) = h(u) \cdot h(v) = h© for all u*v = c in G. ::For example log(a*b) = log(a) + log(b) :::Since log is a homomorphism that has an inverse that is also a homomorphism, log is an of groups. :See also and A has morphisms with more than one source object. A f(v_1,\ldots,v_n) = W : : f\colon V_1 \times \cdots \times V_n \to W\text{,} has a corresponding Linear map: F(v_1\otimes \cdots \otimes v_n) = W : : F\colon V_1 \otimes \cdots \otimes V_n \to W\text{,} Back to top Numerical methods :From Wikipedia:Numerical analysis :See also: One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the , since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control arising from the use of arithmetic. solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points? is very similar to interpolation, except that now we want to find the value of the unknown function at a point which is outside the given points. is also similar, but it takes into account that the data is imprecise. Given some points, and a measurement of the value of some function at these points (with an error), we want to determine the unknown function. The -method is one popular way to achieve this. Much effort has been put in the development of methods for solving . :Standard direct methods, i.e., methods that use some :: , , for symmetric (or hermitian) and positive-definite matrix, and for non-square matrices. : :: , , and are usually preferred for large systems. General iterative methods can be developed using a . are used to solve nonlinear equations. If the function is differentiable and the derivative is known, then Newton's method is a popular choice. is another technique for solving nonlinear equations. Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some . Differential equation: If you set up 100 fans to blow air from one end of the room to the other and then you drop a feather into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the for solving an ordinary differential equation. Back to top Generalization of addition and multiplication :Main articles: Algebraic structure, Abstract algebra, and Addition and multiplication can be generalized in so many ways that mathematicians have created a whole system of categories just to organize them. Back to top Set theory "See also: , Set theory, , , Set, and \varnothing is the empty set (the additive identity) \mathbf{U} is the universe of all elements (the multiplicative identity) a \in A means that is a element (or member) of set . In other words a is in A. : \{ x \in \mathbf{A} : x \notin \mathbb{R} \} means the set of all x's that are members of the set such that x is not a member of the reals. Could also be written \{ \mathbf{A} - \mathbb{R} \} A set does not allow multiple instances of an element. \{1,1,2\} = \{1,2\} :A multiset does allow multiple instances of an element. \{1,1,2\} \neq \{1,2\} A set can contain other sets. \{1,\{2\},3\} \neq \{1,2,3\} A \subset B means that is a proper subset of : A \subseteq A means that is a subset of itself. But a set is not a proper subset of itself. A \cup B is the Union of the sets and . In other words, \{A+B\} : \{1,2\}+\{2,3\}=\{1,2,3\} A \cap B is the Intersection of the sets and . In other words, \{A \bullet B\} All a's in B. :Associative: A \bullet \{B \bullet C\} = \{A \bullet B\} \bullet C :Distributive: A \bullet \{B + C\}=\{A \bullet B\} + \{A \bullet C\} :Commutative: \{A \bullet B\} =\{B \bullet A\} A \setminus B is the Set difference of and . In other words, \{A - A \bullet B\} : \overline{A} or A^c = \{U - A\} is the complement of A. A \bigtriangleup B or A \ominus B is the Anti-intersection of sets and which is the set of all objects that are a members of either or but not in both. : A \bigtriangleup B = (A + B) - (A \bullet B) = (A - A \bullet B) + (B - A \bullet B) A \times B is the Cartesian product of and which is the set whose members are all possible ordered pairs where is a member of and is a member of . The Power set of a set is the set whose members are all of the possible subsets of . A of a set X is a collection of sets whose union contains X as a subset.Wikipedia:Cover (topology) A subset A of a topological space X is called (in X) if every point x in X either belongs to A or is arbitrarily "close" to a member of A. :A subset A of X is if it can be expressed as the union of countably many nowhere dense subsets of X. of sets A_0 = {1, 2, 3} and A_1 = {1, 2, 3} can be computed by finding: : \begin{align} A^*_0 & = \{(1, 0), (2, 0), (3, 0)\} \\ A^*_1 & = \{(1, 1), (2, 1), (3, 1)\} \end{align} so : A_0 \sqcup A_1 = A^*_0 \cup A^*_1 = \{(1, 0), (2, 0), (3, 0), (1, 1), (2, 1), (3, 1)\} Let H'' be the subgroup of the integers (''mZ', +) = ({..., −2''m, −''m'', 0, m'', 2''m, ...}, +) where m'' is a positive integer. :Then the of ''H are the ''mZ' + a'' = {..., −2''m+''a'', −''m''+''a'', a'', ''m+''a'', 2''m''+''a'', ...}. :There are no more than m'' cosets, because ''mZ''' + m'' = ''m('''Z + 1) = m'Z'. :The coset (m'Z' + a'', +) is the congruence class of ''a modulo m''.Joshi p. 323 :Cosets are not usually themselves subgroups of G, only subsets. \exists means "there exists at least one" \exists! means "there exists one and only one" \forall means "for all" \land means "and" (not to be confused with wedge product) \lor means "or" (not to be confused with antiwedge product) Back to top Probability \vert A \vert is the cardinality of A which is the number of elements in A. See measure. P(A) = {\vert A \vert \over \vert U \vert} is the unconditional probability that A will happen. P(A \mid B) = {\vert A \bullet B \vert \over \vert B \vert} is the conditional probability that A will happen given that B has happened. P(A + B) = P(A) + P(B) - P(A \bullet B) means that the probability that ''A or B'' will happen is the probability of ''A plus the probability of B'' minus the probability that both ''A and B'' will happen. P(A \bullet B) = P(A \bullet B \mid B)P(B) = P(A \bullet B \mid A)P(A) means that the probability that ''A and B'' will happen is the probability of "A and B given B" times the probability of B. P(A \bullet B \mid B) = \frac{P(A \bullet B \mid A) \, P(A)}{P(B)}, is permutation relates to the act of '''arranging' all the members of a set into some sequence or . The number of permutations of n'' distinct objects is [[factorial|''n!]].Wikipedia:Permutation :a derangement is a permutation of the elements of a set, such that no element appears in its original position. In other words, derangement is a permutation that has no . The number of of a set of size n'', usually written [[subfactorial|!''n]], is called the "derangement number" or "de Montmort number".Wikipedia:derangement ::The are a triangular array of integers that enumerate permutations of the set { 1, ..., n'' } with specified numbers of fixed points: in other words, '''partial derangements'.Wikipedia:rencontres numbers a combination is a selection of items from a collection, such that the order of selection does not matter. For example, given three numbers, say 1, 2, and 3, there are three ways to choose two from this set of three: 12, 13, and 23. More formally, a k''-'''combination' of a set S'' is a subset of ''k distinct elements of S''. If the set has ''n elements, the number of k''-combinations is equal to the binomial coefficient : \binom nk = \textstyle\frac{n!}{k!(n-k)!}. Pronounced n choose k. The set of all ''k-combinations of a set S'' is often denoted by \textstyle\binom Sk . The '''central limit theorem' (CLT) establishes that, in most situations, when are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed.Wikipedia:Central limit theorem (or bell-shaped curve) where each band has a width of 1 standard deviation – See also: ]] In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.Wikipedia:standard deviation The is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure. :In contrast, the describes the probability of k successes in n draws with replacement.Wikipedia:Hypergeometric distribution See also and Back to top Tactical thinking :From Wikipedia:Game theory :See also Wikipedia:Strategy (game theory) In the accompanying example there are two players; Player one (blue) chooses the row and player two (red) chooses the column. Each player must choose without knowing what the other player has chosen. The payoffs are provided in the interior. The first number is the payoff received by Player 1; the second is the payoff for Player 2. Tit for tat is a simple and highly effective tactic in game theory for the iterated prisoner's dilemma. An agent using this tactic will first cooperate, then subsequently replicate an opponent's previous action. If the opponent previously was cooperative, the agent is cooperative. If not, the agent is not.Wikipedia:Tit for tat In zero-sum games the sum of the payoffs is always zero (meaning that a player can only benefit at the expense of others). Cooperation is impossible in a zero-sum game. John Forbes Nash proved that there is a Nash equilibrium (an optimum tactic) for every finite game. In the zero-sum game shown to the right the optimum tactic for player 1 is to randomly choose A or B with equal probability. Strategic thinking differs from tactical thinking by taking into account how the short term goals and therefore optimum tactics change over time. For example the opening, middlegame, and endgame of chess require radically different tactics. Back to top Rules of Reasoning in Philosophy :From Wikipedia:Philosophiæ Naturalis Principia Mathematica In the four rules, as they came finally to stand in the 1726 edition, Newton effectively offers a methodology for handling unknown phenomena in nature and reaching towards explanations for them. Rule 1: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. Rule 2: Therefore to the same natural effects we must, as far as possible, assign the same causes. Rule 3: The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever. Rule 4: In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, not withstanding any contrary hypothesis that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions. Classical mechanics :Newtonian mechanics, Lagrangian mechanics, and Hamiltonian mechanics Quantum mechanics Highly recommend: Thinking Physics Is Gedanken Physics by Lewis Carroll Epstein Back to top Dimensional analysis :From Wikipedia:Dimensional analysis :See The dimension of a physical quantity can be expressed as a product of the basic physical dimensions length (L), mass (M), time (T), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J), each raised to a rational power. Only commensurable quantities (physical quantities having the same dimension) may be compared, equated, added, or subtracted. Any physical law that accurately describes the real world must be independent of the units (e.g. km or mm) used to measure the physical variables. Consequently, every possible commensurate equation for the physics of the system can be written in the form : a_0 \cdot D_0 = (a_1 \cdot D_1)^{p_1} (a_2 \cdot D_2)^{p_2}...(a_n \cdot D_n)^{p_n} Suppose we wish to calculate the when fired with a vertical velocity component V_\mathrm{y} and a horizontal velocity component V_\mathrm{x} , assuming it is fired on a flat surface. The quantities of interest and their dimensions are then : as L : V_\mathrm{x} as L /T : V_\mathrm{y} as L /T : as L /T2 The equation for the range may be written: : range = (V_x)^a (V_y)^b (g)^c Therefore : \mathsf{L}_\mathrm{x} = (\mathsf{L}_\mathrm{x}/\mathsf{T})^a\,(\mathsf{L}_\mathrm{y}/\mathsf{T})^b (\mathsf{L}_\mathrm{y}/\mathsf{T}^2)^c\, and we may solve completely as a=1 , b=1 and c=-1 . Back to top Computers :https://repl.it/languages, https://www.wolframalpha.com :See also: * * * * * * * * * * * * * * * Back to top Search Math wiki type=search External links *http://mathinsight.org *https://math.stackexchange.com References Category:Foundations